4 BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS 337 for all «, where (ak) is a Hilbert basis of E. For each integer N we have there- N fore £ \\un(av)\\2 ^ /? for all n; hence, letting n tend to + oo, we have fc=0 £lKa,)||2:g/J fc=0 for all N; consequently (5.3.1), i;e J^2(E) and \\v\\l £ p. Finally, for each s > 0, there exists n0 such that \\um - un\\2 g e for all m^n0 and all n^nQy and the same argument shows that \\v — un\\2^e for all n^n0. Hence v = lim un in J£?2(E). n-* oo It is easily verified that the Banach algebra &2(E) ™ not a star algebra (Problem 13). Remark (15.4.8.6) Let (an) be a Hilbert basis of E. To each continuous operator u e JSf (E) we may associate the double sequence (cmil) = ((w(an) | bmj) (some- times called the infinite matrix of u with respect to (#„)). We have seen above that, if u is a Hilbert-Schmidt operator, then ||w||2 = ^ \cmn\2- Con- m, n versely, let (cmn) be any double sequence of complex numbers such that the double family (|cmn|2) is summable. Then, for all «, the vector bn = J] cmnam m is defined, and we have \\bn\\2 = ^ \cmn\2 (6.5.2). Furthermore, for each vector x = Z tnan (with ||*||2 = X |^|2\ the vector u(x) = J tnbn is defined, and n n n ||w(x)||2 ^(X Icmnl2>| ' II*II2; f°r Ae Cauchy-Schwarz inequality shows that \m, n ) if h g A: we have n-h from which follows the absolute convergence of the series £ £n bn and the n inequality above for ||M(X)||. We have therefore defined a continuous endo- morphism u of E (5.5.1), and it is immediately verified that ZM*B)ll2 = I>mj2, so that M e JSf 2(E).at lim \\u — un\\2 = 0. Hence, in view of